432 D'ALEMBERT. 



lie followed it up with another first attempted by him, 

 namely, the variation which might occur to the former 

 results, if the earth, instead of being a sphere oblate at the 

 poles, were an elliptic spheroid, whose axes were different. 

 He added an investigation of the Precession on the sup- 

 position of the form being any other curve approaching 

 the circle. This is an investigation of as great diffi- 

 culty perhaps as ever engaged the attention of analysts. 

 It remains to add that Euler, in 1750, entered on the 

 same inquiries concerning Precession and Nutation ; 

 and with his wonted candour, he declared that he had 

 read D'Alembert's memoir before he began the investi- 

 gation. 



The only other works of D'Alernbert which it is neces- 

 sary to mention, are his three papers on the integral cal- 

 culus. Of these one, in the Berlin Memoirs, is replete with 

 improvements extremely important in the methods of in- 

 tegration, and contains a method of treating linear equa- 

 tions of any order that serve as a foundation for the 

 approximate solutions, which are absolutely indispensable 

 to physical astronomy in the present imperfect state of 

 the calculus. The other two are in the French Academy's 

 Memoirs for 1757 and 1769, the latter giving the demon- 

 strations of the theorems on integration contained in the 

 former. It is in the twenty-first of these that he claims 

 having suggested, as we have already seen, to Clairaut his 

 equation of conditions in the case of three variables. The 

 ' Opuscules' contain likewise, especially the 4th, 5th, and 

 7th volumes, some most important papers on the calculus. 

 Nor must we omit to record that there is every reason 

 to give him credit for having discovered Taylor's Theorem. 

 It is certain that he first gave this celebrated formula 



